Is it obligatory to pay the real value of an unknown owner's item that was sold for a low price, and donate the selling price in charity?

You must repent to Allah for selling the lost item (the necklace) without publicizing it for a year, as publicizing it was obligatory. If you know its owner and he agrees to the sale and to donating the proceeds as charity, there is no problem. If he agrees to the sale but not to donating the proceeds as charity, then pay him the price. If he does not agree to the sale and the necklace is no longer available, then its owner is entitled to a similar item or its value. If you do not know the owner of the necklace, then donate the difference between the selling price and its true value as charity.

As for the stolen helmet, donate the difference between the price you sold it for and its true value as charity, as long as you do not know its owner. If you find him in the future, he has the choice between the reward of the charity or compensation for the value of the helmet, and you may seek recourse from the seller. The value is the usual price, not necessarily the price at which it was sold.